To find non-trivial solutions of (REF ), we apply a Nash-Moser scheme. At each step, we shall find an approximate right inverse, with nice tame estimates, of the linearized operator \(d_{i,\kappa }{F}(i_0)\)  at the reversible torus \(i_0\)  of the current step. Applying the Berti-Bolle theory [1]} and [2]}, one can conjugate the linearized operator \(d_{i,\kappa }{F}(i_0)\)  by a suitable diffeomorphism of the toroidal phase space \(\mathbb {T}^d\times \mathbb {R}^d\times L_{\perp }^{2}\)  in order to obtain a triangular system in the action-angle-normal variables up to nice error terms. Then to solve the triangular system, it is sufficient to find an almost approximate right inverse for the linearized operator in the normal directions denoted \({L}_{\perp }\)  and related to the linearized operator \(\mathcal {L}_{\varepsilon r}\)  of (REF ) through the relation (). To do so, we conjugate \({L}_{\perp }\)  to a constant coefficients operator up to error terms. This is the content of the Section . First, in Proposition  we use a KAM reduction process with quasi-periodic symplectic change of variables to reduce the transport part of \(\mathcal {L}_{\varepsilon r}.\)  Then, in Proposition REF  we look at the localization effects in the normal directions to recover the reduction of the transport part for \({L}_{\perp }.\)  This provides a diagonal operator of order 1 plus a remainder of order \((0,-1)\)  in the variables \((\varphi ,\theta ).\)  Finally, in Proposition REF  we use a KAM reduction process in the Toeplitz operators class to reduce the remainder term. Each KAM reduction occurs for suitable restrictions of the parameters \((\alpha ,\omega )\)  to a Cantor-like set. In addition, the inversion of the final diagonal operator also requires an extraction of parameters. The Nash-Moser process constructs a non-trivial solution
\((\alpha ,\omega )\mapsto (i_\infty (\alpha ,\omega ),\kappa _{\infty }(\alpha ,\omega ))\)  with reversible torus \(i_{\infty }\)  modulo restriction of the parameters to a Cantor set in \((\alpha ,\omega )\)  obtained gathering all the restrictions of all the steps for the contruction of the almost approximate right inverses of the linearized operators. Then, coming back to (REF ), we rigidify the frequency \(\omega \)  into \(\omega (\alpha ,\varepsilon )\)  so that
\(\kappa _{\infty }\big (\alpha ,\omega (\alpha ,\varepsilon )\big )=-\omega _{\textnormal {Eq}}(\alpha ).\) 
